----- When first discussing [[primary ideal|primary ideals]] of a [[commutative ring|commutative]] [[ring]] $A$, we observed: - That the primary ideals of $\mathbb{Z}$ were precisely the powers of prime ideals; - That the primary decomposition of an ideal of $\mathbb{Z}$ is [[Fundamental Theorem of Arithmetic|unique]], and this implies every ideal is a product of prime powers in a unique way These facts hold more generally for [[Dedekind domain|Dedekind domains]]. > [!proposition] Proposition. ([[primary decomposition in Dedekind domains behaves like prime factorization of integers]]) Let $A$ be a [[Dedekind domain]] and $(0) \neq \mathfrak{p} \in \text{Spec }A$. > > 1. The set of $\mathfrak{p}$-[[primary ideal|primary]] [[ideal|ideals]] of $A$ is $\{ \mathfrak{p}^{n} \}_{n \geq 1}$, and $\mathfrak{p}^{n+1} \subsetneq \mathfrak{p}^{n}$. > 2. Let $\mathfrak{a}$ be a nonzero [[ideal]] of $A$. Then $\mathfrak{a}=\mathfrak{p}_{1}^{e_{1}} \cdots \mathfrak{p}_{n}^{e_{n}}=\mathfrak{p}_{1}^{e_{1}} \cap \dots \cap \mathfrak{p}_{n}^{e_{n}},$ > where $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n} \in \text{Spec}(A)-\{ 0 \}$ are distinct and $e_{i} \geq 1$. Furthermore, $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ and $e_{1},\dots,e_{n}$ are uniquely determined (up to reordering of the factors). ^proposition > [!proof]- Proof. ([[primary decomposition in Dedekind domains behaves like prime factorization of integers]]) > Notes. **1.** Notes. **2.** Let $\mathfrak{a} \neq (0)$ be an [[ideal]] of a [[Dedekind domain]] $A$. Then $\mathfrak{a}$ has a [[primary ideal|minimal primary decomposition]] since $A$ is [[Noetherian ring|Noetherian]]. In light of **(1)**, this can be written as $\mathfrak{a}=\mathfrak{p}_{1}^{e_{1}} \cap \dots \cap \mathfrak{p}_{n}^{e_{n}} \ (*)$ for $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ nonzero (since $\mathfrak{a} \neq (0)$) and [[prime ideal|prime]]. The [[uniqueness properties of primary decomposition|associated prime ideals]] $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ are in $\text{mSpec }A$ (nonzero + $A$ has [[Krull dimension|dimension of a ring]] $1$), so must each be isolated as there are no proper inclusions among them. So $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ are the isolated prime ideals of $A$, and $\mathfrak{p}_{1}^{e_{1}},\dots,\mathfrak{p}_{n}^{e_{n}}$ are the isolated primary components of $\mathfrak{a}$. So $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ and $\mathfrak{p}_{1}^{e_{1}},\dots,\mathfrak{p}_{n}^{e_{n}}$ are uniquely determined by $\mathfrak{}$\mathfrak{a}$. Hence $e_{1},\dots,e_{n}$ are also uniquely determined by $\mathfrak{a}$, cf. the 'strictly decreasing' part of claim **(1)**. Thus, the decomposition $(*)$ is unique. We claim it equals the product $\mathfrak{p}_{1}^{e_{1}} \cdots \mathfrak{p}_{n}^{e_{n}}$. Indeed, because $\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}$ are pairwise maximal, they are [[relatively prime integers|coprime]]; a small exercise shows $\mathfrak{p}_{1}^{e_{1}},\dots,\mathfrak{p}_{n}^{e_{n}}$ are then also coprime. Thus $\mathfrak{p}_{1}^{e_{1}} \cap \dots \cap \mathfrak{p}_{n}^{e_{n}}=\mathfrak{p}_{1}^{e_{1}}\cdots \mathfrak{p}_{n}^{e_{n}}$ by the [[Chinese remainder theorem|CRT]]. $ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```